Exceptional Bannai-Ito polynomials

TL;DR

This paper introduces a new class of 1-step exceptional Bannai-Ito polynomials constructed via a generalized Darboux transformation, expanding the theory of discrete orthogonal polynomials with explicit eigenfunctions.

Contribution

It presents the first construction of 1-step exceptional Bannai-Ito polynomials using a generalized Darboux transformation and identifies eight classes of gauge factors involved.

Findings

Constructed 1-step exceptional Bannai-Ito polynomials with discrete orthogonality.

Derived eigenfunctions expressed as 3x3 determinants.

Identified eight classes of gauge factors for the construction.

Abstract

We construct a non-trivial type of 1-step exceptional Bannai-Ito polynomials which satisfy discrete orthogonality by using a generalized Darboux transformation. In this generalization, the Darboux transformed Bannai-Ito operator is directly obtained through an intertwining relation. Moreover, the seed solution, which consists of a gauge factor and a polynomial part, plays an important role in the construction of these 1-step exceptional Bannai-Ito polynomials. And we show that there are 8 classes of gauge factors. We also provide the eigenfunctions of the corresponding multiple-step exceptional Bannai-Ito operator which can be expressed as a 3 x 3 determinant.